Properties

Label 2-150-1.1-c5-0-15
Degree $2$
Conductor $150$
Sign $-1$
Analytic cond. $24.0575$
Root an. cond. $4.90485$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s − 9·3-s + 16·4-s − 36·6-s − 4·7-s + 64·8-s + 81·9-s − 500·11-s − 144·12-s − 288·13-s − 16·14-s + 256·16-s + 1.51e3·17-s + 324·18-s − 1.34e3·19-s + 36·21-s − 2.00e3·22-s − 4.10e3·23-s − 576·24-s − 1.15e3·26-s − 729·27-s − 64·28-s − 2.64e3·29-s − 5.61e3·31-s + 1.02e3·32-s + 4.50e3·33-s + 6.06e3·34-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.0308·7-s + 0.353·8-s + 1/3·9-s − 1.24·11-s − 0.288·12-s − 0.472·13-s − 0.0218·14-s + 1/4·16-s + 1.27·17-s + 0.235·18-s − 0.854·19-s + 0.0178·21-s − 0.880·22-s − 1.61·23-s − 0.204·24-s − 0.334·26-s − 0.192·27-s − 0.0154·28-s − 0.584·29-s − 1.04·31-s + 0.176·32-s + 0.719·33-s + 0.899·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(24.0575\)
Root analytic conductor: \(4.90485\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: $\chi_{150} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 150,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p^{2} T \)
3 \( 1 + p^{2} T \)
5 \( 1 \)
good7 \( 1 + 4 T + p^{5} T^{2} \)
11 \( 1 + 500 T + p^{5} T^{2} \)
13 \( 1 + 288 T + p^{5} T^{2} \)
17 \( 1 - 1516 T + p^{5} T^{2} \)
19 \( 1 + 1344 T + p^{5} T^{2} \)
23 \( 1 + 4100 T + p^{5} T^{2} \)
29 \( 1 + 2646 T + p^{5} T^{2} \)
31 \( 1 + 5612 T + p^{5} T^{2} \)
37 \( 1 + 7288 T + p^{5} T^{2} \)
41 \( 1 + 18986 T + p^{5} T^{2} \)
43 \( 1 + 2404 T + p^{5} T^{2} \)
47 \( 1 - 8900 T + p^{5} T^{2} \)
53 \( 1 - 39804 T + p^{5} T^{2} \)
59 \( 1 + 28300 T + p^{5} T^{2} \)
61 \( 1 - 18290 T + p^{5} T^{2} \)
67 \( 1 - 65956 T + p^{5} T^{2} \)
71 \( 1 + 28800 T + p^{5} T^{2} \)
73 \( 1 + 30808 T + p^{5} T^{2} \)
79 \( 1 - 60228 T + p^{5} T^{2} \)
83 \( 1 + 2468 T + p^{5} T^{2} \)
89 \( 1 - 22678 T + p^{5} T^{2} \)
97 \( 1 + 36968 T + p^{5} T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.84396098943132870838267193099, −10.62570855440475318875704801535, −9.959872337523174859360159898778, −8.183032768116416497077553074852, −7.17019351791619532055988830999, −5.83779549707084058758380350803, −5.05795382503485700523026063621, −3.63637193428282476954024249578, −2.04602721607580926517974810890, 0, 2.04602721607580926517974810890, 3.63637193428282476954024249578, 5.05795382503485700523026063621, 5.83779549707084058758380350803, 7.17019351791619532055988830999, 8.183032768116416497077553074852, 9.959872337523174859360159898778, 10.62570855440475318875704801535, 11.84396098943132870838267193099

Graph of the $Z$-function along the critical line